Fiberwise Langlands duality for the Hitchin fibration

Let $\mathcal A$ be the Hitchin base, let $H:M\to\mathcal A$ be the Hitchin fibration, and let $\overline P$ be the universal Poincaré sheaf from the global conjecture. For each $\underline a\in\mathcal A$, write $H^{-1}(\underline a)$ for the Hitchin fiber and let $\overline P_{\underline a}:=\overline P|_{H^{-1}(\underline a)}$. Fiberwise Langlands duality conjecture. For every $\underline a\in\mathcal A$, the Fourier–Mukai transform with kernel $\overline P_{\underline a}$,

$$\Phi^{\overline P_{\underline a}}:D^b(H^{-1}(\underline a))\to D^b(H^{-1}(\underline a)),$$

is an auto-equivalence of the bounded derived category $D^b(H^{-1}(\underline a))$ of coherent sheaves on $H^{-1}(\underline a)$.

This is the fiberwise consequence expected from the global autoduality conjecture. The source gives no resolution of the assertion for arbitrary, possibly singular Hitchin fibers, so it remains open.